We use exactly the same idea as when constructing the integers:
consider pairs (a,b) where $b\neq{}0.$ (The idea is that this
means a/b.) Say that (a,b)=(c,d) exactly when ad=bc. (That is:
say that they are "equivalent" when this holds, and then work with
equivalence classes of pairs instead of just with pairs.)

The rules for doing arithmetic with these are exactly the ones
you learned in school for doing arithmetic with fractions. For
instance, (a,b)+(c,d)=(ad+bc,bd).